Quasitriangular Structures for a Class of Hopf Algebras of Dimension p⁶

Abstract

We study quasitriangular structures in the A ⋆ H which is the twisted smash product Hopf algebra. As an application, we consider a class of non-commutative and non-cocommutative Hopf algebras H _p (α, β, q, m) which are pointed non semisimple, and establish quasitriangular structures on H _p (α, β, q, m). Another objective of this paper is to study the ring extensions A ^biH  ⊂ A ⊂ A ⋆ H. We prove a Maschke-type theorem for an A ⋆ H-modules. We form an associated Morita context [A ^biH , A, A, A ⋆ H] and use these to investigate some of properties of the Doi–Takeuchi double and the Drinfeld's double.

Key Words:

• Twisted smash product
• Hopf algebra
• Mashke-type theorem
• Morita context

Acknowledgments

The authors are very grateful to referee for bringing other References (Beattie et al., 2000; Caenepeel and Dăscălescu, 1998; Larson and Radford, 1995; Nenciu, 2001) to their attention and for many valuable comments and corrections which have improved some definitions, results and the conditions in the first, the second and the third versions of this manuscripts. The first author also thank his colleagues and friends of the Department of Mathematics in Chonbuk National University: In-Soo Kim, Yong Hun Lee, Gyeongsig Seo, Jae Up So, and Cho YongHwan, and the whole people of the first author's family. Their warm hospitality created a good atmosphere. In particular, we also thank Professor Hong Jae Lee for his help with the post-doctoral fellowship heartily. This work was supported by the grant of Post-Doctoral Program, Chonbuk National University (2002). The first author would like to thank the Chonbuk National University for its warm hospitality. This work was also supported in part by the Science Foundation of Henan Province for Distinguished Young Scholars.

Notes

^#Communicated by M. Cohen.